The choice between brackets (square brackets) and parentheses (round brackets) in math often depends on whether the endpoint of an interval is included or excluded.
Intervals and Endpoints
The key distinction lies in how you want to represent the boundaries of a set of numbers. We use intervals to represent sets of numbers.
- Parentheses ( ): Indicate that the endpoint is not included in the interval. This means the interval approaches the endpoint but does not actually reach it.
- Brackets [ ]: Indicate that the endpoint is included in the interval. The interval includes the endpoint itself.
According to the provided reference, "square brackets mean the end point is included, and round parentheses mean it's excluded."
Open vs. Closed Intervals
Based on whether the endpoints are included or excluded, we can classify intervals:
| Interval Type | Left Endpoint | Right Endpoint | Notation Example | Meaning |
|---|---|---|---|---|
| Open | Excluded | Excluded | (a, b) | All numbers between a and b, but not a or b. |
| Closed | Included | Included | [a, b] | All numbers between a and b, including a and b. |
| Half-Open/Half-Closed | Included | Excluded | [a, b) | All numbers between a and b, including a, but not b. |
| Half-Open/Half-Closed | Excluded | Included | (a, b] | All numbers between a and b, not including a, but including b. |
Examples
- The interval (2, 5) represents all numbers between 2 and 5, not including 2 and 5.
- The interval [2, 5] represents all numbers between 2 and 5, including 2 and 5.
- The interval (2, 5] represents all numbers between 2 and 5, not including 2 but including 5.
- The interval [2, 5) represents all numbers between 2 and 5, including 2 but not including 5.
Infinity
When an interval extends to infinity (either positive or negative), we always use parentheses because infinity is not a number and cannot be included.
- For example, (2, ∞) represents all numbers greater than 2, but not including 2.
- Similarly, (-∞, 5] represents all numbers less than or equal to 5.
Other Uses in Math
Parentheses are also used for:
- Order of Operations: To group terms and define the order in which operations should be performed (e.g., (2 + 3) * 4).
- Function Arguments: To enclose the arguments passed to a function (e.g., sin(x)).
- Coordinates: To represent coordinates in a plane (e.g., (x, y)).
Brackets can also be used for:
- Matrices: To enclose elements of a matrix.
- Floor and Ceiling Functions: Although less common, brackets can denote floor or ceiling functions.
